In this article we formulate the direct and inverse scattering theory
for the focusing matrix Zakharov-Shabat system as the construction of
a 1, 1-correspondence between focusing potentials with entries in $L^1(\mathbb{R})$ and
Marchenko integral kernels, given the fact that these kernels encode the usual
scattering data (one reflection coecient, the discrete eigenvalues with positive
imaginary part, and the corresponding norming constants) faithfully. In the
re
ectionless case, we solve the Marchenko equations explicitly using matrix
triplets and obtain focusing matrix NLS solutions in closed form.